7752
Ind. Eng. Chem. Res. 2006, 45, 7752-7759
KINETICS, CATALYSIS, AND REACTION ENGINEERING Model-Fitting Methods for Evaluation of the Kinetics Triplet during Thermal Decomposition of Poly(ethylene terephthalate) (PET) Soft Drink Bottles Biswanath Saha and Aloke K. Ghoshal* Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati-39, Assam, India
Different chemical reaction models (first-order, second-order, and nth-order) and six different model-fitting techniques (nth-order, Friedman, Freeman-Carroll, Chang, ASTM E689, and the standard deviation minimization technique (SDMT)) are used to evaluate the kinetics triplet (E, ln k0, and n) for the thermal decomposition of waste soft drink poly(ethylene terephthalate) (PET) bottles (M/s Coca Cola). Nonisothermal experiments at four different heating rates (5, 10, 15, and 25 K/min) and isothermal experiments at four different temperatures (685, 693, 703, and 711 K) are conducted using thermogravimetric analysis (TGA). The experimental results are predicted through simulation of the kinetics models of decomposition. The model input parameters (i.e., the optimum kinetics triplet) are used as obtained from the different model-fitting techniques. Results show that the SDMT methodology with Agarwal and Sivasubramaniam approximation for the temperature integral is possibly the best and most versatile method to calculate the optimum kinetics triplet, because it, in contrast to other model-fitting methods, addresses multiple heating rates. The optimum kinetics triplet thus obtained for the thermal decomposition of the waste PET sample is E ) 269.35 kJ/mol, ln k0 ) 44.94, and n ) 1. Introduction The application of a novel alternative process, such as pyrolysis or catalytic decomposition, as a means of reusing scrap tires and waste plastic, has become a subject of renewed interest. In the kinetics study of pyrolysis, it is important to know the decomposition mechanism, the rate of reaction, and the reaction parameters to predict the product distribution. This knowledge helps in proper selection of the reactor, as well as optimization of the reactor design and operating conditions. Thermogravimetic analysis (TGA) coupled with a model-fitting approach is commonly used for such studies to evaluate the apparent overall kinetics parameters, such as activation energy, pre-exponential factor, and the reaction model (kinetics triplet). Literature reports discussed below indicate that different researchers have used different kinetics models and different approximation techniques to evaluate the kinetics triplets for their studies, which include polymer pyrolysis kinetics study and solid reaction kinetics study. This often causes confusion in regard to what model and which technique should be used for evaluation of the kinetics triplet that best represents the system of study. Furthermore, most of the reported studies are based on a single-heating-rate TGA curve, which is not recommended anymore to be used for evaluation of the kinetics triplet. Therefore, through the present work, we have tried to demonstrate a possible way of overcoming such confusion. Pyrolysis study, using TGA coupled with mass spectrometry (TG-MS), of various individual and mixed commodity plastics including poly(ethylene terephthalate) (PET) showed that plastics with different molecular structures decomposed at different temperatures.1-4 Application of a simplified kinetics * To whom correspondence should be addressed. Tel.: +91-03612582251. Fax: +91-0361-2690762. E-mail address:
[email protected].
model resulted in discrepancies in the kinetics parameters obtained for polystyrene (PS) from isothermal and nonisothermal experiments. Therefore, a detailed kinetics scheme was proposed for polystyrene (PS) and polyethylene (PE), taking into consideration the heat-transport effect and the complex reaction mechanism.2,3 The pyrolysis kinetics of various substances, including polyamide-6 (to recover -carprolactam),4 scrap tires using a two-step reaction model and single heating rate,5 electronic packaging materials at high temperature and in an oxidizing atmosphere,6 poly(vinyl chloride),7 and a mixture of PE and PS8 has been reported in the literature. The nonisothermal TGA kinetics of poly(trimethyl terephthalate) (PTT) and PET in argon, air, and nitrogen atmospheres by the FreemanCarroll, Friedman, and Chang methods to estimate the kinetics triplet9 and PET thermal decomposition under strict pyrolysis condition and with different proportions of oxygen using TGA10 were studied. Thermal degradation of the polymer is mostly described by a random scission mechanism, which does not follow first-order kinetics rigorously. Therefore, a relationship for determining the reaction order was established to avoid the blind use of first-order kinetics.11 In our previous paper, we used the nth-order technique and the ASTM E698 technique to determine the apparent kinetics triplet for the thermal degradation of PET using TGA.12 Nonisothermal kinetics for the polymer was reported by Mamleev et al.13 In another paper, Mamleev et al.14 reported cotton decomposition via modulated thermogravimetry (MTG), using single or multiple heating rates and multistep reaction mechanisms. They reported that, in many cases, it is possible that the obtained kinetics triplet describes the rate-limiting step of the decomposition process and recommended that this approach is acceptable for chemical engineering applications. The artificial neural network technique15 and various computational methods16 were also applied to study
10.1021/ie060282x CCC: $33.50 © 2006 American Chemical Society Published on Web 10/13/2006
Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006 7753 Table 1. Experimental Conditions for TGA Studies Non-isothermal Experimental Conditions sample
initial mass (mg)
heating rate (K min-1)
temperature range (K)
% residue
Tw0 (K)
Td (K)
Tm (K)
Tw∞ (K)
waste PET waste PET waste PET waste PET virgin PET virgin PET virgin PET virgin PET
7.962 8.255 9.452 9.605 16.74 14.59 14.63 14.034
5 10 15 25 5 10 15 25
303-873 303-873 303-873 303-873 303-873 303-873 303-873 303-873
10.05 12.18 13.35 12.72 12.17 12.40 12.21 12.205
622.01 625.51 623.08 623.82 619.18 623.32 621.48 621.59
639.39 655.43 664.57 671.76 642.98 653.33 662.95 670.23
703.03 715.05 723.59 737.81 708.99 719.47 728.197 741.12
90.42 786.64 788.69 792.12 788.86 789.69 792.43 794.52
Isothermal Experimental Conditions sample
initial mass (mg)
sample temperature (K)
weight loss before reaching the target temperature (%)
total experimental time (min)
% residue
waste PET waste PET waste PET waste PET virgin PET virgin PET virgin PET virgin PET
45.9978 52.4871 42.1638 29.6361 14.64 16.53 13.92 15.83
685 693 703 711 687 699 708 718
1.9 1.49 1.78 4.087 20.43 19.81 14.75 47.28
0-84.3 0-94.4 0-94.4 0-94.6 0-94.25 0-84.15 0-64.33 0-64.67
16.85 17.06 16.03 13.59 13.07 13.51 12.51 12.11
thermal decomposition kinetics. Thus, the traditional modelfitting kinetics analysis using a single-heating-rate and singlestep decomposition model gives only a single set of kinetics triplet, which is estimated after minimizing the deviation between simulated data and experimental data. However, presently, the International Confederation of Thermal Analysis and Calorimetry (ICTAC) project, in 2000, ruled out the validity of thermal kinetics analysis using a single heating rate.17 Modern model-fitting thermal kinetics analysis methods use multiple heating rates, address multistep reactions, and incorporate possible partial diffusion, back reactions, branch reactions, etc. in the model equations.17,18 Still, the selection of an appropriate model and the initial guess of kinetics parameters are major drawbacks of the model-fitting method.18 In this paper, we used different chemical reaction models and six model-fitting techniques to estimate the kinetics triplet of waste PET decomposition using single and multiple heating rates to select the best realistic kinetics analysis technique and the best reaction model by minimizing the deviation between theoretical and experimental data. The decomposition of waste PET sample under both nonisothermal and isothermal conditions is compared with that of a virgin PET sample. We proposed and applied a new and versatile approach based on standard deviation minimization technique (SDMT), which can use both single and multiple heating rates to investigate nonisothermal decomposition kinetics. Thus, SDMT overcomes the uncertainties associated with the traditional single-heating-rate modelfitting techniques. Moreover, it is based on single-parameter optimization, which overcomes the possible uncertainties and complexities associated with the multiparameter optimization technique.13,18 Thus, SDMT has inherent advantages over the nth-order technique discussed in our earlier paper,12 where threeparameter optimization was used. This method is also used to obtain the kinetics triplet for isothermal decomposition. The obtained kinetics triplet is used to simulate the nonisothermal and isothermal kinetics data. The simulated data are compared with experimental data for both conditions to select the best reaction kinetics model and kinetics analysis technique. We also applied model-fitting methods such as nth-order, FreemanCarroll, Friedman, Chang, and ASTM E698 techniques, which use a single heating rate and found that they predicted the experimental data well. However, kinetics analysis using several TGA curves with varying heating rates, using SDMT, gives us
the scope of avoiding the so-called compensation effect14,16,19 for one heating rate TGA curve, because of the unambiguous choice of order of the chemical reaction models. SDMT involves minimization of the square of the average standard deviations between the experimental and computed data using single- and multiple-heating-rate TGA curves to evaluate the optimum values of kinetics parameters. The Agrawal and Sivasubramanian20 approximation is used for integration of the temperature integral. The proposed SDMT is more accurate, because it better predicts the experimental data of isothermal and nonisothermal degradation of the PET sample, in comparison to most of the traditional single-heating-rate model-fitting techniques. Experimental Procedure and Equipment The nonisothermal and isothermal decompositions were conducted with individual waste PET soft drinks bottles (M/s Coca Cola) and virgin PET sample (AS-40, bottle grade) that were supplied by South Asian Petrochem Limited, India. Details of experimental procedure are illustrated elsewhere.12,21 Experiments were conducted in a TGA instrument (Mettler Toledo, Model No. TGA/SDTA 851e) under a nitrogen atmosphere for a temperature range of 303-873 K. The nitrogen flow rate was maintained at 40-50 mL/min, according to the specifications of the equipment. PET samples were shredded into very small pieces (mesh size of -40/+60) and fed directly into the TGA instrument. An alumina crucible (with a capacity of 70 µL for the nonisothermal case and 900 µL for the isothermal case) was used as a sample holder. The experiments were repeated three times at a heating rate of 10 K/min, to confirm the repeatability and authenticity of the generated data. Experiments were conducted under dynamic conditions at different heating rates (5, 10, 15, and 25 K/min). The total mass of the sample with the corresponding experimental conditions is given in Table 1. Variations in the conversion (R) and rate of reaction (dR/dT), relative to temperature, during the nonisothermal pyrolysis of waste and virgin PET samples at different heating rates are reported in Figures 1 and 2, respectively. For isothermal experiments, the temperature program was optimized to reach the preset temperature of experiments within maximum of 6.2 min when the sample temperature was regulated within (1 K of the setpoints. Four different temperatures were maintained to study the isothermal decomposition
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Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006 Table 2. Experimental Conditions for Proximate Analysis (ASTM E1131)a temperature range (°C)
heating rate (°C min-1)
time (min)
Non-isothermal Heating 33.33 1.5 16.67 6 92.85 7 20 2.5
50-100 100-200 200-850 850-900
Isothermal Heating 2.5 2.5-10
900 900
environment nitrogen nitrogen nitrogen nitrogen nitrogen oxygen
a The sample mass for both non-isothermal and isothermal heating was 19.208 mg.
Figure 1. Variation of conversion (R) with temperature during nonisothermal pyrolysis of the virgin and waste PET samples.
experiments were repeated three times at a target temperature of 685 K to test the repeatability of the experiments. The proximate analysis (method ASTM E1131)22 of the sample was conducted in a TGA instrument (Mettler Toledo, Model No. TGA/SDTA 851e). The total mass of sample, the temperature program, and the experimental conditions are reported in Table 2. The obtained result can be broadly divided into three steps. The first step (e200 °C) does not show any substantial change in the percentage normalized weight. This step involves the loss of a very small quantity of moisture. The second step (7 min (200 °C)-19.5 min (900 °C)) indicates the maximum loss of volatile and polymer content (∼92%). The third step (19.5-22.5 min) is indicative of the loss of fixed carbon (∼8%). Kinetics Models and Techniques The nth-order kinetics model equation, combined with the Arrhenius approach of the temperature function of reaction rate constant, is
dR E ) k0 exp f(R) dt RT
(
Figure 2. Variation of rate of reaction (dR/dT) with temperature during nonisothermal pyrolysis of the virgin and waste PET samples.
Figure 3. Variation of conversion (R) with time during isothermal pyrolysis of the waste and virgin PET samples.
of both virgin and waste PET samples. Figure 3 represents the variations in the conversion (R), relative to temperature, during isothermal decomposition at different target temperatures. The
)
(1)
where t is time (given in minutes), T the temperature (given in Kelvin), R the conversion of the reaction (R ) (W0 - W)/(W0 - W∞), where W0 is the initial weight of the sample, W the sample weight at any temperature T, and W∞ the final sample weight (all sample weight given in milligrams)), dR/dt the rate of reaction (expressed in units of min-1), and f(R) the reaction model. The terms k0, which represents the pre-exponential factor (expressed in units of K-1), and E, which is the activation energy (expressed in units of kJ/mol), are the Arrhenius parameters. R is the universal gas constant (expressed in units of kJ mol-1 K-1). The reaction model may take various forms, based on nucleation and nucleus growth, the phase boundary reaction, diffusion, and the chemical reaction.12-14,17-19 The reactive gases (such as air and oxygen) have important roles, in regard to selection of the reaction models, in the case of polymer decomposition. In the present case, we have considered chemical reaction models of orders 1, 2, and n. At a constant heating rate under nonisothermal conditions, the explicit temporal/time dependence in eq 1 is eliminated through the trivial transformation
β
( )
Ea dR ) k0 exp f(R) dT RT
(2)
where β is the heating rate (expressed in units of K/min; β ) dT/dt) and dR/dT is the rate of reaction (expressed in units of K-1).
Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006 7755
(1) Model-Fitting Method for Nonisothermal Experiments. Details of the model-fitting techniques, such as the nth-order model technique and the ASTM E698 technique, have already been reported by Saha and Ghoshal12 in a recent publication. Wang et al.9 elaborated on the other model-fitting methods, such as the Freeman-Carroll, Friedman, and Chang techniques. (2) Standard Deviation Minimization Technique (SDMT). A new technique that is almost similar to the method proposed by Mamleev and Bourbigot13 to obtain the kinetics triplet from a single TGA curve by means of minimization of the square of the average standard deviation ∆, between the experimental and computed data, is applied. In this method, the kinetics model described by eq 2 can be written as
g(R) )
k
dR 0 T ) ∫0 ∫0R f(R) β
R
(
exp -
k0 E dT ) I(E,TR) RT β
)
(3)
where
I(ER,TR) )
∫0T
R
( )
exp -
ER dT RT
Figure 4. Dependency of exact and optimal ln(k0/β) on the optimal E obtained from the SDMT method.
In this paper, we concentrated on one-parameter optimization, using the Agrawal and Sivasubramanian20 approximation of integration of the temperature integral I(E,TR). The objective function in the case of a single-heating-rate TGA curve for calculation of the square of the average standard deviation can be written as J
∆(E) )
∑ j)1
[( ) gj(Rj)
-
ln
Ij(E,TRj)
where
() k0
ln
β
)
avg
1 J
1 J
exp
∑ j)1
( )]
2
gj(Rj)
J
ln
Ij(E,TRj)
(4)
cal
( )
gj(R) )
gj(Rj)
J
ln ∑ I (E,T j)1 j
(5)
Rj)
In the case of multiple-heating-rate TGA curves, eqs 4 and 5 are deduced to calculate optimum values of ∆(E): M
∆(E) )
{ [( J
∑ ∑ ln I m)1 j)1
gj,m(Rj,m)
-
∑ j)1
ln
j,m(E,TRj,m) exp J
( ) ]} ( ) ∑[ ∑ ( )] 1 J
k0
ln
)
β
)
avg
1
M
1
M m)1 J j)1
ln
(8)
The subscript j corresponds to the reaction model that is selected. For each reaction model, the rate constants are evaluated at several temperatures (Ti) and the Arrhenius parameters are determined from the following equation:
ln(kj(Ti)) ) ln(kj) (6)
cal
gj(Rj)
J
dR ) kj(Ti)t ∫0R f(R)
Ej RTi
(9)
2
gj,m(Rj,m)
Ij,m(E,TRj,m)
mization is performed via a numerical method, using MATLAB7. The nonlinear least-squares solver function “lsqnonlin”, without constraint, was applied to solve the problem using a “large-scale trust-region reflective Newton algorithm”. The numerical integration of the temperature integral was performed using the Agrawal and Sivasubramanian20 approximation, because it has already been proven to be superior to the other two approximations (Coats and Redfern, Gorabachev20) for E ) 40-250 kJ/mol over the temperature range of 300-1000 K.20 (4) Model-Fitting Method for Isothermal Experiments.19 Equation 1 can be rearranged as
(7)
Ij(E,TRj)
A range of R values, from 0.1 to 0.9, is used for the evaluation of ∆(E). In the case of SDMT for a single heating rate, the order of reaction (n) is varied from 0.5 to 2.1 using increments of 0.05. The optimal values of E are obtained for different values of n by minimizing ∆(E). Thereafter, average values of ln(k0/β) are calculated from eq 7, using the optimum values of E for each value of n. The optimum values of E and ln(k0/β), together with the optimal magnitude of ∆(E), are represented as function of n in Figure 4. In the case of multiple heating rates, a similar approach was used. No change in the value of n was observed for the minimum value of ∆(E). Therefore, this is not reported in the present paper. (3) Numerical Procedure and Algorithms for Standard Deviation Minimization. The objective function (∆(E)) mini-
Akaike’s Information Criteria (AIC),23 which is discussed below, is applied to choose the appropriate reaction model.
AIC ) N ln
(SSN ) + 2K
(10)
where N is the number of data points, K the number of parameters, and SS the sum of squares of the vertical distance of the points from the center. The corrected AIC value for a small number of data points is obtained from
AICc ) AIC +
2K(K + 1) N-K-1
(11)
Thus, for two reaction models A and B,
∆AIC ) AICC,B - AICC,A
(12)
AICC,B is the corrected AIC for model B and AICC,A is the corrected AIC for model A.
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Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006
Table 3. Kinetics Parameters Using Different Non-isothermal Model-Fitting Techniques ln k0,a single-rate data
sample
method
E (kJ)
n
waste PET waste PET waste PET waste PET waste PET waste PET
ASTM E698 Freeman- Carroll Friedman Chang nth-order model SDMT single heating rate single heating rate multiple heating rates
214.91 288.99 278.56 287.87 322.3
1 1.2085 1.216 1 1.724
35.628 ( 0.0241 48.649 ( 0.0140 46.973 ( 0.0199 47.903 ( 0.0425 54.76 ( 0.0137
272.77 384.63 269.35
1 2 1
45.614 ( 0.008794 65.472 ( 0.01931
a
β ) 5 °C/min
45.21 ( 0.0094
ln k0,a Multiple-Rate Data β ) 10 °C/min β ) 15 °C/min
β ) 25 °C/min
45.02 ( 0.0094
44.69 ( 0.0056
44.85 ( 0.0069
Values of ln k0 also include the standard deviation.
The model with the lower AICc score is the model that is more likely to be correct. Results and Discussion (1) Nonisothermal Decomposition. The present investigation considered the thermal decomposition of waste and virgin PET samples in the first part only (up to T ) 786-790 K), because it is the main pyrolysis step.12 The experiments were conducted at four different heating rates (5, 10, 15, and 25 K/min) in a nitrogen environment. The experiments were repeated three times at a heating rate of 10 K/min, to test the repeatability, and the experiments showed almost similar behaviors during pyrolysis. This pyrolysis step, which yields a weight loss of 85%-90%, starts at a temperature near Td and finishes at a temperature near TW∞. The temperature at which R ) 0 (Tw0), the temperature at which decomposition starts (Td), the temperature at which the maximum weight loss rate occurs (Tm), and the temperature at the end of the pyrolysis step (Tw∞) are reported in Table 1 for each case of experiments. The initial weight of the sample is taken at temperature Tw0, to eliminate the moisture and volatile compound content. A quick thermal decomposition is observed in the range of Td to Tw∞ (see Figure 1) and the highest decomposition rate is observed at ∼Tm (see Figure 2), as reported in Table 1. After this quick increase, the solid continues to decompose smoothly and slowly until the end of the experiment. The thermal decomposition behavior of both the virgin and waste PET samples are almost similar, except for a slight difference in Tm (3-5 K). The single peak that is observed in Figure 2 does not necessarily indicate a single-step reaction that is occurring during pyrolysis. However, for the simplicity of kinetics analysis, it is assumed to be a single-step reaction5,16 subsequently. Table 1 shows that the residue that remained for waste PET sample is in the range of 10%-13% and that for the virgin PET sample is ∼12%. (2) Isothermal Decomposition. It is evident from the nonisothermal experiment (Table 1) that waste PET thermal decomposition starts in the temperature range of 639-672 K and the maximum decomposition temperature lies in the range of 703-738 K, depending on the heating rate. Also, at a heating rate of 10 K/min, which is the universal heating rate, the maximum decomposition temperature is 715 K. Therefore, the four different temperaturessviz., 685, 693, 703, and 711 Ks of the isothermal experiments were chosen to be between 672 K and 715 K (i.e., between the start of decomposition and the maximum decomposition temperature) with an approximate interval of 10 K. The thermal decomposition behavior of the virgin PET varies significantly from that of waste PET samples (see Figure 3). Holland and Hay24 have already reported that the isothermal decomposition temperature for virgin PET is 643-683 K. Therefore, the observed variation in the present study can be attributed to the fact that, in contrast to waste PET
sample, very fast decomposition occurs in the case of the virgin PET samples while reaching the target temperature (685-718 K). The percentage loss of waste and virgin PET samples before reaching the target temperature under different conditions is reported in Table 1. For each isothermal experiment with waste PET samples, a weight loss of 82%-85% is observed within 62.5-67 min after reaching the preset isothermal temperature (6-6.2 min). After this quick decrease, the solid continues to decompose smoothly and very slowly to the end of the experiment.21 We have taken W0 at t0 when the preset temperature is reached and W∞ at t∞ when the weight loss is ∼99.5% of the total weight loss for the calculation of R.21 Table 1 also indicates that the residue left for the waste PET sample is within 13%-17% and that for the virgin PET sample is within 12%14%. Holland and Hay25 reported residues of 18%-20% for different PET samples under isothermal conditions (643-683 K). The marginally lower value of residue in the present work is possibly due to study at relatively higher temperatures (685718 K). (3) Kinetics for Nonisothermal and Isothermal ModelFitting Analysis. (a) Nonisothermal Model-Fitting Analysis. A single-step reaction is assumed during nonisothermal decomposition of the PET sample, and a one-step chemical reaction model is used to calculate kinetics parameters from TGA curves, using different techniques, as discussed previously. It may be observed from Table 3 that the calculated values of kinetics parameters such as E, ln k0, and n via the nth-order model and ASTM E698 techniques are the highest and the lowest, respectively. The Freeman-Carroll, Friedman, and Chang methods give more-or-less constant values for these parameters (Table 3). These can be attributed to the fact that different calculating techniques approximate the thermal decomposition behaviors in different temperature ranges. The nth-order model technique uses a wide temperature range, starting from Tw0 to Tw∞. The ASTM E698 method uses only Tm, which is much lower than Tw∞. The kinetics triplet reported here using the ASTM E698 method differs significantly from our previous report.12 This may be attributed to the use of different heating rates (5, 10, and 15 K/min in the present case and 10, 15, and 25 K/min in the previous case). In fact, this is the limitation of the ASTM E698 method, because, in this method, one point for each heating rate is taken for calculation of the kinetics triplet. Furthermore, according to the ASTM E698 method,22 the heating rate should be preferably within 20 °C. In the present paper, the Freeman-Carroll and Chang techniques describe the behavior of thermal decomposition in the temperature range from (Tm - [30-60]) to Tm. However, the E value given by the Friedman technique is somewhat lower, mainly because the thermal decomposition behavior is estimated in the lower temperature range from [Td - (20-40)] to Td than other techniques. However, the standard deviation values are much
Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006 7757 Table 4. Present Work and Literature Reported Kinetics Parameters kinetics parameter E (kJ/mol)
sample waste PET sample from beverage bottles waste PET sample from beverage bottles waste PET sample from beverage bottles waste PET sample from beverage bottles waste PET sample from beverage bottles virgin PET sample virgin PET sample virgin PET sample virgin PET I sample waste PET sample from beverage bottles virgin PET I sample virgin PET E and PET I sample virgin PET sample
Isothermal Experiment 196.96 1 220 ( 10 1 200 ( 10 1 220 ( 10 214 ( 2 1.15
31.81 29.89 25.996 29.29 15.2 ( 0.04
( )
kj,0 + ln(I(Ej,Tm)) ) C β
(13)
Applying the Agrawal and Sivasubramanian approximation, eq 13 can be written as
()
{ [ ( ) ]}
k0 RTm2 ln ) C - ln β E
2RTm E RTm 2 1-5 E
ln k0 (min)
18.00 43.077 30.297
less using SDMT (Table 3) for both the single-heating-rate and multiple-heating-rate curves. It was also observed that the standard deviation is minimum for n ) 1 using SDMT, when a single heating rate is concerned. Therefore, n ) 1 is also considered for multiple heating rates. Unfortunately, in the case of only one kinetics curve (one heating rate), the calculated optimum E and ln(k0/β) are completely determined by the model that is used and can greatly differ for different models for all traditional model-fitting kinetics analysis techniques.13,14 This circumstance requires special investigation. Because the curve (Figure 2) is almost symmetrical around the maximum decomposition temperature (Tm), to illustrate this behavior, let us rewrite eq 3 for the maximum decomposition temperature (Tm) as follows:
ln(g(RTm)) ) ln
n
Non-isothermal Experiment 322.3 1.724 162.15 1 256.4 1.05 271.6 1.09 269.35 1 242 1 238.7 1.15 259.34 1 220 ( 10 1
1-
+
E (14) RTm
The term ln(RTm2/E){(1 - 2RTm/E)/[1 - 5(RTm/E)2]} and the C values vary slightly with different reaction models and can be considered to be constants. The values of optimal E and ln(k0/β) vary greatly with the shift in the order of the reaction, which is called the compensation effect.12,13,19 The term ln(k0/β) that is obtained from eq 14, corresponding to the optimal E, is termed here as exact. The dependency of the exact and optimal ln(k0/β) on the optimal E is shown in Figure 4. Figure reflects that the first-order model can be selected as the best model. Thus, this is similar to the observation by the SDMT method with the single-heating-rate curve, using the Agrawal and Sivasubramanian approximation discussed previously. Therefore, the best kinetics triplet is obtainable with the firstorder model and the corresponding optimal values of E and ln k0. However, Tm changes with different heating rates and ln k0 becomes dependent on Tm. Therefore, different optimal kinetics triplets are possible for different heating rates. Therefore, to obtain a unique kinetics triplet for thermal decomposition of the waste PET sample, the SDMT method using multipleheating-rate TGA curves can be applied, because only this technique, among the other model-fitting methods, has the
54.76 26.37 36.88 41.98 44.94
reference 12 (nth-order method) 12 (ASTM E698 method) 10 (for main pyrolysis step) 26 (for main pyrolysis step) present work 27 28 present work 24 present work 24 25 28
feature of using multiple heating rates. However, the difference in the optimum values of E and ln k0 obtained by single- and multiple-heating-rate TGA curves in the present case is not significant for the first-order model (see Table 3). (b) Isothermal Model-Fitting Techniques. The model-fitting technique is also applied for isothermal kinetics analysis to obtain the kinetics triplet. Values of E and ln k0 obtained using the model-fitting method for isothermal kinetics analyses are 196.96 kJ/mol and 28.515, respectively, for the first-order model and 210.25 kJ/mol and 31.81, respectively, for the second-order model. Furthermore, Akaike’s criterion analysis, as discussed previously, shows that the first-order model is more likely. It was observed that, for all four temperatures of study, the AICc value is lower for the first-order model. Thus, similar to nonisothermal study, the isothermal study also indicates in preference of the first-order model reaction of thermal decomposition of the waste PET sample. The kinetics parameters obtained in the present work and from the literature, both under isothermal and nonisothermal conditions, are reported in Table 4. It is observed from the table that the author’s result in the present work are in close agreement with the literature reported data. Numerical Simulation (1) Nonisothermal Data Prediction. The nth-order kinetics model equation with initial conditions of R ) 0 at T ) 630 K is solved numerically via the Runge-Kutta 4th-order method, using the kinetics parameters obtained by nth-order model techniques and Freeman-Carroll, Friedman, Chang, and ASTM E698 techniques. On the other hand, the nth-order kinetics model equation also can be solved after transforming eq 3 as follows:
R)1-
[( )
] ]
k0 I(E,TR)(n + 1) + 1 β
[()
R ) 1 - exp -
k0 I(E,TR) β
1/(n-1)
(for n * 1) (15)
(for n ) 1)
(16)
The kinetics parameters obtained from SDTM techniques are used to simulate nonisothermal decomposition data using eqs 15 and 16. It is observed that, although all the techniques predicted fairly the experiment, the SDTM and nth-order model techniques better predict the nonisothermal experimental data, which is evident from Figures 5 and 6. This is further supported by the statistical analysis results, which have been reported in the form of standard deviations (see Table 3).
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Figure 5. Percentage deviation of the simulation results for nonisothermal decomposition at a heating rate of 10 K/min. Figure 7. Isothermal (685 K) prediction from various model-fitting kinetics analysis techniques at a heating rate of 10 K/min.
Figure 6. Simulations and the experimental data for nonisothermal decomposition (waste PET) using SDMT (multiple heating rates) for the first-order model.
(2) Isothermal Data Prediction. The kinetics triplet that was determined by several model fitting techniques from nonisothermal experiments is used for isothermal predictions. Figure 7 represents the isothermal prediction at 685 K by various model-fitting kinetics analysis techniques at a heating rate of 10 K/min. The figure indicates that the Friedman and ASTM E698 techniques predict the values well, compared to the other techniques. It is also interesting to observe that SDMT (for both the cases of a single heating rate and multiple heating rates) well-predicted the experimental data. Therefore, it may be concluded that SDMT is likely to be the most versatile method for predicting the kinetics triplet for thermal decomposition of the waste PET sample. Isothermal prediction of experimental data at 711 K, also from SDMT, is reported in Figure 8. The deviations observed in the figures may be attributed to the fact that PET degradation is a complex phenomenon, which involves many reaction steps. Broken values of n, as reflected in Figure 4, also possibly indicated overlapping of the reaction steps. Thus, the degradation description, in terms of a standard power-law model, is not sufficient to cover wide operative conditions. However, the authors envisage further study, using multistep reactions to justify such behavior in the future research plan.
Figure 8. Isothermal (711 K) prediction from the SDMT method (multiple heating rates).
Conclusion The following conclusions can be drawn from the isothermal and nonisothermal kinetics study of the waste poly(ethylene terephthalate) (PET) sample. (1) The proposed new and versatile approach based on the standard deviation minimization technique (SDMT) can effectively use both single and multiple heating rates to investigate nonisothermal decomposition kinetics. It is based on single parameter optimization and overcomes the possible uncertainties and complexities that are associated with the multiparameter optimization technique. (2) Among the model-fitting techniques used, the nth-order model technique and SDMT better predicts the experimental data of the nonisothermal degradation of the PET sample. However, SDMT is the most versatile and probably the best method, because it takes multiple heating rates into account. (3) SDMT also predicted the isothermal decomposition of the PET sample well. However, the degradation description, in terms of a standard power-law model, may not be sufficient to cover the wide operative conditions and requires investigation
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using multistep reactions. Both the isothermal and nonisothermal studies resulted in a common conclusion that the pyrolysis of the PET sample studied is best-represented by a first-order chemical reaction model. (4) The optimum values of the kinetics triplet for the thermal degradation of the PET sample are E ) 269.35 kJ/mol, ln k0 ) 44.94, and n ) 1. Literature Cited (1) Bockhorn, H.; Hornung, A.; Hornung, U. Environmental engineering: Stepwise pyrolysis of plastic waste. Chem. Eng. Sci. 1999, 54, 3043. (2) Bockhorn, H.; Hornung, A.; Hornung, U.; Schawaller, D. Kinetic study on the thermal degradation of polypropylene and polyethylene. J. Anal. Appl. Pyrolysis 1999, 48, 93. (3) Bockhorn, H.; Hornung, U.; Hornung, A.; Jakobstroer, P. Modelling of isothermal and dynamic pyrolysis of plastics considering nonhomogeneous temperature distribution and detailed degradation mechanism. J. Anal. Appl. Pyrolysis 1999, 49, 53. (4) Bockhorn, H.; Donner, S.; Gernsbeck, M.; Hornung, A.; Hornung, U. Pyrolysis of polyamide 6 under catalytic conditions and its application to reutilization of carpets. J. Anal. Appl. Pyrolysis 2001, 58-59, 79. (5) Leung, D. Y. C.; Wang, C. L. Kinetics study of scrap tyre pyrolysis and combustion. J. Anal. Appl. Pyrolysis 1998, 45, 153. (6) Liou, T. Kinetics study of thermal decomposition of electronic packaging material. Chem. Eng. J. 2004, 98, 39. (7) Marongiu, A.; Faravelli, T.; Bozzano, G.; Colombo, M.; Dente, M.; Ranzi, E. Thermal degradation of poly(vinyl chloride). J. Anal. Appl. Pyrolysis 2003, 70, 519. (8) Faravelli, T.; Bozzano, G.; Colombo, M.; Ranzi, E.; Dente, M. Kinetic modeling of thermal degradation of polyethylene and polystyrene mixtures. J. Anal. Appl. Pyrolysis 2003, 70, 761. (9) Wang, X.; Li, X.; Yan, D. Thermal decomposition kinetics of poly(trimethylene terephthalate). Polym. Degrad. Stab. 2000, 69, 361. (10) Martin-Gullon, I.; Esperanza, M.; Font, R. Kinetic model for the pyrolysis and combustion of poly-(ethylene terephthalate) (PET). J. Anal. Appl. Pyrolysis 2001, 58-59, 635. (11) Gao, Z.; Amasaki, I.; Nakada, M. A thermogravimetric study on thermal degradation of polyethylene. J. Anal. Appl. Pyrolysis 2003, 67, 1. (12) Saha, B.; Ghoshal, A. K. Thermal degradation kinetics of poly(ethylene terephthalate) from waste soft drinks bottles. Chem. Eng. J. 2005, 111, 39. (13) Mamleev, V.; Bourbigot, S.; Le Bras, M.; Duquesne, S.; Sˇ esta´k, J. Modelling of nonisothermal kinetics in thermogravimetry. Phys. Chem. Chem. Phys. 2000, 2, 4708. (14) Mamleev, V.; Bourbigot, S. Modulated thermogravimetry in analysis of decomposition kinetics. Chem. Eng. Sci. 2005, 60, 747.
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ReceiVed for reView March 9, 2006 ReVised manuscript receiVed July 11, 2006 Accepted September 7, 2006 IE060282X
